Macro-criticality from micro-critical agents

Research summary

One paper, summarised for reference. Emergent Macro-Criticality from Micro-Critical Agents — Nicolas Bessone & Erwan Plantec, arXiv:2605.01818 (2026). IT University of Copenhagen / Barcelona Computational Foundation. arXiv:2605.01818 · code.

The question

Does tuning each individual agent near its own dynamical critical point make the collective critical? The paper builds a two-scale system to test exactly this cross-scale propagation of criticality — and finds that near-critical internal dynamics are not sufficient for collective critical-like avalanches. What matters instead is whether activity can propagate through the interaction network.

The model

• Micro (reservoir). Each agent holds a discrete-time stochastic recurrent network of binary neurons with branching-process dynamics: an active neuron activates each other with homogeneous probability $p^{\text{micro}}$, so the nominal critical point is $p^{\text{micro}}_c = 1/(N_{\text{neurons}}-1)$ (one activation per active neuron on average) — subcritical activity dies, near-critical self-sustains, supercritical saturates.
• Macro (environment). $N_{\text{agents}}=256$ light-signalling agents on a 2D torus. Each senses the angular distribution of lit neighbours within a vision radius $v_r$ (discretised into $E$ sectors), feeds that into its reservoir’s input neurons, and a readout neuron sets its own light on/off — a static spatial sensing graph driving a dynamic activity network.
• Avalanches are extracted from the population light activity $A_l(t)=\sum_i l_i(t)$ (size $S=\sum_t A_l$, duration $T$), and scored against the branching exponents via $L = |\alpha_S-1.5| + |\alpha_T-2| + \mathrm{KS}_S + \mathrm{KS}_T$ (lower = closer to critical branching).

Micro-criticality is not sufficient

The macro criticality landscape $L(p^{\text{micro}}, v_r)$ is a structured phase diagram — silence at low $p^{\text{micro}}$/$v_r$, saturation at high, and a narrow curved band of near-critical avalanches between — and that band is displaced from the reservoir’s own critical point $p^{\text{micro}}_c$, shifting systematically with $v_r$. So collective near-criticality lives in a restricted region set by the spatial interactions, not inherited from tuning the agents.

Connectivity controls it

Activity propagation is a network property. The interaction graph is a random geometric graph; its giant connected component appears at a connectivity threshold $v_r^{\text{conn}} = \sqrt{L^2\ln N_{\text{agents}} / ((N_{\text{agents}}-1)\pi)} \approx 47$ for their parameters. Below it the graph is fragmented and activity stays local; above it, activity spreads population-wide. As $v_r$ (hence mean degree) grows, the effective branching ratio $\sigma_{\text{eff}}$ reaches 1 at progressively lower $p^{\text{micro}}$ — spatial coupling substitutes for internal propagation. Recast in average degree $\kappa$, the same structure holds: critical-like behaviour is sustained across an extended range of connectivity, not a single point.

Two findings worth flagging

• Slightly subcritical micro is more robust. The width of the near-critical band in $\kappa$ depends on $p^{\text{micro}}$: values just below $p^{\text{micro}}_c$ support near-criticality across a broader range of macroscopic connectivity than critical or supercritical ones — a candidate reason some biological networks (e.g. gene-regulatory) sit subcritically when embedded in larger interaction networks.
• Exponents deviate from mean-field. At the best operating point the size exponent is $\alpha_S\approx1.7$ — consistently above the mean-field $3/2$ even though the score $L$ rewards $1.5$ — with $\alpha_T\approx2.0$. The authors attribute the shift to constrained, spatially structured propagation (violating homogeneous mixing) and finite size, echoing other reservoir/spatial systems.

Why it’s collected here

This is the closest published analog to this lab’s central question: agents whose controller is a reservoir (micro/network scale) coupled through a spatial sensing graph into a collective (macro/ensemble scale), asking whether network criticality propagates to ensemble criticality. Its answer — it does not, by itself; the coupling/propagation structure is decisive, the two critical points do not coincide, and slightly-subcritical internal dynamics are the most robust — is the same conclusion the forage-criticality work has been converging on, reached with the same branching/avalanche + Clauset power-law machinery (and the same $\alpha_S\approx1.7$ deviation).

References

Bessone & Plantec (2026), arXiv:2605.01818 · Beggs & Plenz (2003), J. Neurosci. 23, 11167 · Kinouchi & Copelli (2006), Nat. Phys. 2, 348 · Mora & Bialek (2011), J. Stat. Phys. 144, 268 · Clauset, Shalizi & Newman (2009), SIAM Rev. 51, 661 · Cramer et al. (2020), Nat. Commun. 11, 2853 · Pontes-Filho, Nichele & Lepperød (2025), arXiv:2508.02218 · Romanczuk & Daniels (2022), Order, Disorder and Criticality · Torres-Sosa, Huang & Aldana (2012), PLoS Comput. Biol. 8, e1002669.